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      <p class="mytitle">2D density plot</p>
      <p style="color: #d0d0d0">
        <a href="#definition">definition</a> - <a href="#mistake">mistake</a> -
        <a href="#related">related</a> -
        <a href="#code">code</a>
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              <div id="definition" class="section level1 mycontent">
<h1>Definition</h1>
<hr />
<p>This page is dedicated to a group of graphics allowing to study the
<code>combined distribution</code> of two quantitative variables. These
graphics are basically extensions of the well known <a
href="https://www.data-to-viz.com/graph/density.html">density plot</a>
and <a
href="https://www.data-to-viz.com/graph/histogram.html">histogram</a>.
<br><br> The global concept is the same for each variation. One variable
is represented on the X axis, the other on the Y axis, like for a <a
href="https://www.data-to-viz.com/graph/scatter">scatterplot</a>
(<code>1</code>). Then, the number of observations within a particular
area of the 2D space is counted and represented by a
<code>color gradient</code>. The shape can vary:</p>
<ul>
<li>Hexagones are often used, leading to a hexbin chart
(<code>2</code>)</li>
<li>Squares make 2d histograms (<code>3</code>)</li>
<li>It is also possible to compute kernel density estimate to get 2d
density plots (<code>5</code>) or contour plots (<code>6</code>)</li>
</ul>
<p><br> Here is an overview of these different possibilities</p>
<div class="sourceCode" id="cb1"><pre
class="sourceCode python"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" tabindex="-1"></a><span class="co"># Libraries</span></span>
<span id="cb1-2"><a href="#cb1-2" tabindex="-1"></a><span class="im">import</span> numpy <span class="im">as</span> np</span>
<span id="cb1-3"><a href="#cb1-3" tabindex="-1"></a><span class="im">import</span> matplotlib.pyplot <span class="im">as</span> plt</span>
<span id="cb1-4"><a href="#cb1-4" tabindex="-1"></a><span class="im">from</span> scipy.stats <span class="im">import</span> gaussian_kde <span class="im">as</span> kde</span>
<span id="cb1-5"><a href="#cb1-5" tabindex="-1"></a></span>
<span id="cb1-6"><a href="#cb1-6" tabindex="-1"></a><span class="co"># Create data: 200 points</span></span>
<span id="cb1-7"><a href="#cb1-7" tabindex="-1"></a>data <span class="op">=</span> np.random.multivariate_normal([<span class="dv">0</span>, <span class="dv">0</span>], [[<span class="dv">1</span>, <span class="fl">0.5</span>], [<span class="fl">0.5</span>, <span class="dv">3</span>]], <span class="dv">200</span>)</span>
<span id="cb1-8"><a href="#cb1-8" tabindex="-1"></a>x, y <span class="op">=</span> data.T</span>
<span id="cb1-9"><a href="#cb1-9" tabindex="-1"></a></span>
<span id="cb1-10"><a href="#cb1-10" tabindex="-1"></a><span class="co"># Create a figure with 6 plot areas</span></span>
<span id="cb1-11"><a href="#cb1-11" tabindex="-1"></a>fig, axes <span class="op">=</span> plt.subplots(ncols<span class="op">=</span><span class="dv">6</span>, nrows<span class="op">=</span><span class="dv">1</span>, figsize<span class="op">=</span>(<span class="dv">21</span>, <span class="dv">5</span>))</span>
<span id="cb1-12"><a href="#cb1-12" tabindex="-1"></a></span>
<span id="cb1-13"><a href="#cb1-13" tabindex="-1"></a><span class="co"># Everything starts with a Scatterplot</span></span>
<span id="cb1-14"><a href="#cb1-14" tabindex="-1"></a>axes[<span class="dv">0</span>].set_title(<span class="st">&#39;Scatterplot&#39;</span>)</span>
<span id="cb1-15"><a href="#cb1-15" tabindex="-1"></a>axes[<span class="dv">0</span>].plot(x, y, <span class="st">&#39;ko&#39;</span>)</span>
<span id="cb1-16"><a href="#cb1-16" tabindex="-1"></a></span>
<span id="cb1-17"><a href="#cb1-17" tabindex="-1"></a><span class="co"># Thus we can cut the plotting window in several hexbins</span></span>
<span id="cb1-18"><a href="#cb1-18" tabindex="-1"></a>nbins <span class="op">=</span> <span class="dv">20</span></span>
<span id="cb1-19"><a href="#cb1-19" tabindex="-1"></a>axes[<span class="dv">1</span>].set_title(<span class="st">&#39;Hexbin&#39;</span>)</span>
<span id="cb1-20"><a href="#cb1-20" tabindex="-1"></a>axes[<span class="dv">1</span>].hexbin(x, y, gridsize<span class="op">=</span>nbins, cmap<span class="op">=</span>plt.cm.BuGn_r)</span>
<span id="cb1-21"><a href="#cb1-21" tabindex="-1"></a></span>
<span id="cb1-22"><a href="#cb1-22" tabindex="-1"></a><span class="co"># 2D Histogram</span></span>
<span id="cb1-23"><a href="#cb1-23" tabindex="-1"></a>axes[<span class="dv">2</span>].set_title(<span class="st">&#39;2D Histogram&#39;</span>)</span>
<span id="cb1-24"><a href="#cb1-24" tabindex="-1"></a>axes[<span class="dv">2</span>].hist2d(x, y, bins<span class="op">=</span>nbins, cmap<span class="op">=</span>plt.cm.BuGn_r)</span>
<span id="cb1-25"><a href="#cb1-25" tabindex="-1"></a></span>
<span id="cb1-26"><a href="#cb1-26" tabindex="-1"></a><span class="co"># Evaluate a gaussian kde on a regular grid of nbins x nbins over data extents</span></span>
<span id="cb1-27"><a href="#cb1-27" tabindex="-1"></a>k <span class="op">=</span> kde(data.T)</span>
<span id="cb1-28"><a href="#cb1-28" tabindex="-1"></a>xi, yi <span class="op">=</span> np.mgrid[x.<span class="bu">min</span>():x.<span class="bu">max</span>():nbins<span class="op">*</span><span class="ot">1j</span>, y.<span class="bu">min</span>():y.<span class="bu">max</span>():nbins<span class="op">*</span><span class="ot">1j</span>]</span>
<span id="cb1-29"><a href="#cb1-29" tabindex="-1"></a>zi <span class="op">=</span> k(np.vstack([xi.flatten(), yi.flatten()]))</span>
<span id="cb1-30"><a href="#cb1-30" tabindex="-1"></a></span>
<span id="cb1-31"><a href="#cb1-31" tabindex="-1"></a><span class="co"># plot a density</span></span>
<span id="cb1-32"><a href="#cb1-32" tabindex="-1"></a>axes[<span class="dv">3</span>].set_title(<span class="st">&#39;Calculate Gaussian KDE&#39;</span>)</span>
<span id="cb1-33"><a href="#cb1-33" tabindex="-1"></a>axes[<span class="dv">3</span>].pcolormesh(xi, yi, zi.reshape(xi.shape), cmap<span class="op">=</span>plt.cm.BuGn_r)</span>
<span id="cb1-34"><a href="#cb1-34" tabindex="-1"></a></span>
<span id="cb1-35"><a href="#cb1-35" tabindex="-1"></a><span class="co"># add shading</span></span>
<span id="cb1-36"><a href="#cb1-36" tabindex="-1"></a>axes[<span class="dv">4</span>].set_title(<span class="st">&#39;2D Density with shading&#39;</span>)</span>
<span id="cb1-37"><a href="#cb1-37" tabindex="-1"></a>axes[<span class="dv">4</span>].pcolormesh(xi, yi, zi.reshape(xi.shape), shading<span class="op">=</span><span class="st">&#39;gouraud&#39;</span>, cmap<span class="op">=</span>plt.cm.BuGn_r)</span>
<span id="cb1-38"><a href="#cb1-38" tabindex="-1"></a></span>
<span id="cb1-39"><a href="#cb1-39" tabindex="-1"></a><span class="co"># contour</span></span>
<span id="cb1-40"><a href="#cb1-40" tabindex="-1"></a>axes[<span class="dv">5</span>].set_title(<span class="st">&#39;Contour&#39;</span>)</span>
<span id="cb1-41"><a href="#cb1-41" tabindex="-1"></a>axes[<span class="dv">5</span>].pcolormesh(xi, yi, zi.reshape(xi.shape), shading<span class="op">=</span><span class="st">&#39;gouraud&#39;</span>, cmap<span class="op">=</span>plt.cm.BuGn_r)</span>
<span id="cb1-42"><a href="#cb1-42" tabindex="-1"></a>axes[<span class="dv">5</span>].contour(xi, yi, zi.reshape(xi.shape) )</span></code></pre></div>
<pre><code>## &lt;matplotlib.contour.QuadContourSet object at 0x16502df10&gt;</code></pre>
<div class="sourceCode" id="cb3"><pre
class="sourceCode python"><code class="sourceCode python"><span id="cb3-1"><a href="#cb3-1" tabindex="-1"></a><span class="co"># save</span></span>
<span id="cb3-2"><a href="#cb3-2" tabindex="-1"></a>plt.savefig(<span class="st">&quot;IMG/density2d.png&quot;</span>)</span>
<span id="cb3-3"><a href="#cb3-3" tabindex="-1"></a>plt.show()</span></code></pre></div>
<p><img src="density2d_files/figure-html/unnamed-chunk-2-1.png" width="2016" /></p>
</div>
<p><br><br></p>
<div class="mycontent">
<div id="what-for" class="section level1">
<h1>What for</h1>
<hr />
<p>2d distribution are very useful to avoid <a
href="http://www.data-to-viz.com/caveat/overplotting.html">overplotting</a>
in a <a
href="https://www.data-to-viz.com/graph/scatter">scatterplot</a>. Here
is an example showing the difference between an overplotted scatterplot
and a 2d density plot. In the second case, a very obvious hidden pattern
appears:</p>
<div class="sourceCode" id="cb4"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb4-1"><a href="#cb4-1" tabindex="-1"></a><span class="co"># Libraries</span></span>
<span id="cb4-2"><a href="#cb4-2" tabindex="-1"></a><span class="fu">library</span>(tidyverse)</span>
<span id="cb4-3"><a href="#cb4-3" tabindex="-1"></a><span class="fu">library</span>(hrbrthemes)</span>
<span id="cb4-4"><a href="#cb4-4" tabindex="-1"></a><span class="fu">library</span>(viridis)</span>
<span id="cb4-5"><a href="#cb4-5" tabindex="-1"></a><span class="fu">library</span>(patchwork)</span>
<span id="cb4-6"><a href="#cb4-6" tabindex="-1"></a></span>
<span id="cb4-7"><a href="#cb4-7" tabindex="-1"></a><span class="co"># Dataset:</span></span>
<span id="cb4-8"><a href="#cb4-8" tabindex="-1"></a>a <span class="ot">&lt;-</span> <span class="fu">data.frame</span>( <span class="at">x=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="dv">10</span>, <span class="fl">1.2</span>), <span class="at">y=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="dv">10</span>, <span class="fl">1.2</span>), <span class="at">group=</span><span class="fu">rep</span>(<span class="st">&quot;A&quot;</span>,<span class="dv">20000</span>))</span>
<span id="cb4-9"><a href="#cb4-9" tabindex="-1"></a>b <span class="ot">&lt;-</span> <span class="fu">data.frame</span>( <span class="at">x=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="fl">14.5</span>, <span class="fl">1.2</span>), <span class="at">y=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="fl">14.5</span>, <span class="fl">1.2</span>), <span class="at">group=</span><span class="fu">rep</span>(<span class="st">&quot;B&quot;</span>,<span class="dv">20000</span>))</span>
<span id="cb4-10"><a href="#cb4-10" tabindex="-1"></a>c <span class="ot">&lt;-</span> <span class="fu">data.frame</span>( <span class="at">x=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="fl">9.5</span>, <span class="fl">1.5</span>), <span class="at">y=</span><span class="fu">rnorm</span>(<span class="dv">20000</span>, <span class="fl">15.5</span>, <span class="fl">1.5</span>), <span class="at">group=</span><span class="fu">rep</span>(<span class="st">&quot;C&quot;</span>,<span class="dv">20000</span>))</span>
<span id="cb4-11"><a href="#cb4-11" tabindex="-1"></a>data <span class="ot">&lt;-</span> <span class="fu">do.call</span>(rbind, <span class="fu">list</span>(a,b,c))</span>
<span id="cb4-12"><a href="#cb4-12" tabindex="-1"></a></span>
<span id="cb4-13"><a href="#cb4-13" tabindex="-1"></a>p1 <span class="ot">&lt;-</span> data <span class="sc">%&gt;%</span></span>
<span id="cb4-14"><a href="#cb4-14" tabindex="-1"></a>  <span class="fu">ggplot</span>( <span class="fu">aes</span>(<span class="at">x=</span>x, <span class="at">y=</span>y)) <span class="sc">+</span></span>
<span id="cb4-15"><a href="#cb4-15" tabindex="-1"></a>    <span class="fu">geom_point</span>(<span class="at">color=</span><span class="st">&quot;#69b3a2&quot;</span>, <span class="at">size=</span><span class="dv">2</span>) <span class="sc">+</span></span>
<span id="cb4-16"><a href="#cb4-16" tabindex="-1"></a>    <span class="fu">theme_ipsum</span>() <span class="sc">+</span></span>
<span id="cb4-17"><a href="#cb4-17" tabindex="-1"></a>    <span class="fu">theme</span>(</span>
<span id="cb4-18"><a href="#cb4-18" tabindex="-1"></a>      <span class="at">legend.position=</span><span class="st">&quot;none&quot;</span></span>
<span id="cb4-19"><a href="#cb4-19" tabindex="-1"></a>    )</span>
<span id="cb4-20"><a href="#cb4-20" tabindex="-1"></a></span>
<span id="cb4-21"><a href="#cb4-21" tabindex="-1"></a>p2 <span class="ot">&lt;-</span> <span class="fu">ggplot</span>(data, <span class="fu">aes</span>(<span class="at">x=</span>x, <span class="at">y=</span>y) ) <span class="sc">+</span></span>
<span id="cb4-22"><a href="#cb4-22" tabindex="-1"></a>  <span class="fu">stat_density_2d</span>(<span class="fu">aes</span>(<span class="at">fill =</span> ..density..), <span class="at">geom =</span> <span class="st">&quot;raster&quot;</span>, <span class="at">contour =</span> <span class="cn">FALSE</span>) <span class="sc">+</span></span>
<span id="cb4-23"><a href="#cb4-23" tabindex="-1"></a>  <span class="fu">scale_x_continuous</span>(<span class="at">expand =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>)) <span class="sc">+</span></span>
<span id="cb4-24"><a href="#cb4-24" tabindex="-1"></a>  <span class="fu">scale_y_continuous</span>(<span class="at">expand =</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="dv">0</span>)) <span class="sc">+</span></span>
<span id="cb4-25"><a href="#cb4-25" tabindex="-1"></a>  <span class="fu">scale_fill_viridis</span>() <span class="sc">+</span></span>
<span id="cb4-26"><a href="#cb4-26" tabindex="-1"></a>  <span class="fu">theme</span>(</span>
<span id="cb4-27"><a href="#cb4-27" tabindex="-1"></a>    <span class="at">legend.position=</span><span class="st">&#39;none&#39;</span></span>
<span id="cb4-28"><a href="#cb4-28" tabindex="-1"></a>  )</span>
<span id="cb4-29"><a href="#cb4-29" tabindex="-1"></a></span>
<span id="cb4-30"><a href="#cb4-30" tabindex="-1"></a>p1 <span class="sc">+</span> p2</span></code></pre></div>
<p><img src="density2d_files/figure-html/unnamed-chunk-3-3.png" width="960" style="display: block; margin: auto;" /></p>
</div>
<div id="variation" class="section level1">
<h1>Variation</h1>
<hr />
<p>2d distribution is one of the rare cases where using 3d can be worth
it.</p>
<p><br></p>
<p>It is possible to transform the <a
href="https://www.data-to-viz.com/graph/scatter">scatterplot</a>
information in a grid, and count the number of data points on each
position of the grid. Then, instead of representing this number by a
graduating color, the <code>surface plot</code> use 3d to represent
dense are higher than others.</p>
<p><br></p>
<p>In this case, the position of the 3 groups become obvious:</p>
<div class="sourceCode" id="cb5"><pre class="sourceCode r"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" tabindex="-1"></a><span class="fu">library</span>(plotly)</span>
<span id="cb5-2"><a href="#cb5-2" tabindex="-1"></a><span class="fu">library</span>(MASS)</span>
<span id="cb5-3"><a href="#cb5-3" tabindex="-1"></a></span>
<span id="cb5-4"><a href="#cb5-4" tabindex="-1"></a><span class="co"># Compute kde2d</span></span>
<span id="cb5-5"><a href="#cb5-5" tabindex="-1"></a>kd <span class="ot">&lt;-</span> <span class="fu">with</span>(data, MASS<span class="sc">::</span><span class="fu">kde2d</span>(x, y, <span class="at">n =</span> <span class="dv">50</span>))</span>
<span id="cb5-6"><a href="#cb5-6" tabindex="-1"></a></span>
<span id="cb5-7"><a href="#cb5-7" tabindex="-1"></a><span class="co"># Plot with plotly</span></span>
<span id="cb5-8"><a href="#cb5-8" tabindex="-1"></a><span class="fu">plot_ly</span>(<span class="at">x =</span> kd<span class="sc">$</span>x, <span class="at">y =</span> kd<span class="sc">$</span>y, <span class="at">z =</span> kd<span class="sc">$</span>z) <span class="sc">%&gt;%</span> <span class="fu">add_surface</span>()</span></code></pre></div>
<div class="plotly html-widget html-fill-item" id="htmlwidget-f8cf8b7972e0096cc263" style="width:960px;height:576px;"></div>
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</div>
<div id="mistake" class="section level1">
<h1>Common mistakes</h1>
<hr />
<ul>
<li>Play with <a
href="http://www.data-to-viz.com/caveat/bin_size.html">bin size /
bandwidth</a>. As for 2d, density and histograms are really sensitive to
this parameter and can lead to different conclusions.</li>
<li>Useful only if sample size is huge. Use <a
href="https://www.data-to-viz.com/graph/scatter">scatterplot</a> if
there is no overplotting.</li>
<li>Pick up a good <a
href="http://www.data-to-viz.com/caveat/rainbow.html">color
palette</a>.</li>
</ul>
</div>
<div id="related" class="section level1">
<h1>Related</h1>
<hr />
<div class="row">
<div class="col-lg-3 col-md-6 col-sm-6">
<a href="http://www.data-to-viz.com/graph/bubble.html" class="btn btn-primary mybtnrelated" style="margin-bottom:4px;white-space: normal !important;">
<img  src="https://github.com/holtzy/data_to_viz/raw/master/img/section/BubblePlotSmall.png">
<p class="mytitlerelated">
Bubble plot
</p>
<p class="mytextrelated">
Add a third dimension to your scatter: the circle size
</p>
</a>
</div>
<div class="col-lg-3 col-md-6 col-sm-6">
<a href="http://www.data-to-viz.com/graph/correlogram.html" class="btn btn-primary mybtnrelated" style="margin-bottom:4px;white-space: normal !important;">
<img  src="https://github.com/holtzy/data_to_viz/raw/master/img/section/CorrelogramSmall.png">
<p class="mytitlerelated">
Correlogram
</p>
<p class="mytextrelated">
Shows the relationship between each pair of numeric variables
</p>
</a>
</div>
<div class="col-lg-3 col-md-6 col-sm-6">
<a href="http://www.data-to-viz.com/graph/connectedscatter.html" class="btn btn-primary mybtnrelated" style="margin-bottom:4px;white-space: normal !important;">
<img  src="https://github.com/holtzy/data_to_viz/raw/master/img/section/ScatterConnectedSmall.png">
<p class="mytitlerelated">
Connected Scatterplot
</p>
<p class="mytextrelated">
Very close from a scatterplot, but link data points with segments
</p>
</a>
</div>
<div class="col-lg-3 col-md-6 col-sm-6">
<a href="http://www.data-to-viz.com/graph/density2d.html" class="btn btn-primary mybtnrelated" style="margin-bottom:4px;white-space: normal !important;">
<img  src="https://github.com/holtzy/data_to_viz/raw/master/img/section/2dDensitySmall.png">
<p class="mytitlerelated">
Density 2d
</p>
<p class="mytextrelated">
One of the best way to avoid overplotting for big sample size
</p>
</a>
</div>
</div>
</div>
<div id="code" class="section level1">
<h1>Build your own</h1>
<hr />
<p>The <a
href="https://r-graph-gallery.com/2d-density-chart.html">R</a>, <a
href="https:/python-graph-gallery.com/2d-density-plot/">Python</a>, <a
href="https://www.react-graph-gallery.com/2d-density-plot">React</a> and
<a href="https://d3-graph-gallery.com/density2d.html">D3</a> graph
galleries are 4 websites providing hundreds of chart example, always
providing the reproducible code. Click the button below to see how to
build the chart you need with your favorite programing language.</p>
<p>
<a href="https://r-graph-gallery.com/2d-density-chart.html" class="btn btn-primary">R
graph gallery</a>
<a href="https://python-graph-gallery.com/2d-density-plot/" class="btn btn-primary">Python
gallery</a>
<a href="https://www.react-graph-gallery.com/2d-density-plot" class="btn btn-primary">React
gallery</a>
<a href="https://d3-graph-gallery.com/density2d.html" class="btn btn-primary">D3
gallery</a>
</p>
</div>
</div>

          <section
            id="form"
            style="
              margin-top: 70px;
              padding-bottom: 70px;
              padding-top: 70px;
              background-color: #f8f9fa;
            "
          >
            <div class="container">
              <div
                style="
                  display: flex;
                  flex-direction: column;
                  align-items: center;
                "
              >
                <h2 style="font-size: 44px" class="text-center">
                  Dataviz decision tree
                </h2>
                <p
                  class="text-center"
                  style="max-width: 600px; font-size: 18px"
                >
                  <a href="https://www.data-to-viz.com">Data To Viz</a> is a
                  comprehensive <b>classification of chart types</b> organized
                  by data input format. Get a high-resolution version of our
                  decision tree delivered to your inbox now!
                </p>
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                  style="
                    margin-top: -50px;
                    width: 100%;
                    display: flex;
                    justify-content: center;
                  "
                >
                  <script
                    async
                    data-uid="8010f15fd2"
                    src="https://prodigious-trailblazer-3628.ck.page/8010f15fd2/index.js"
                  ></script>
                </div>
                <br />
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                    class="img-fluid"
                    src="../img/poster/poster_small.png"
                    alt="High Resolution Poster"
                    style="max-width: 400px"
                  />
                </div>
              </div>
            </div>
          </section>

           &nbsp;
<hr />
<p style="text-align: center;">A work by <a href="https://www.yan-holtz.com/">Yan Holtz</a> for <a href="https://data-to-viz.com">data-to-viz.com</a></p>

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